Integrand size = 30, antiderivative size = 39 \[ \int \frac {2^{2/3}-2 x}{\left (2^{2/3}+x\right ) \sqrt {-1-x^3}} \, dx=\frac {2\ 2^{2/3} \text {arctanh}\left (\frac {\sqrt {3} \left (1+\sqrt [3]{2} x\right )}{\sqrt {-1-x^3}}\right )}{\sqrt {3}} \] Output:
2/3*2^(2/3)*arctanh(3^(1/2)*(1+2^(1/3)*x)/(-x^3-1)^(1/2))*3^(1/2)
Time = 1.99 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.05 \[ \int \frac {2^{2/3}-2 x}{\left (2^{2/3}+x\right ) \sqrt {-1-x^3}} \, dx=\frac {2\ 2^{2/3} \text {arctanh}\left (\frac {\sqrt {-1-x^3}}{\sqrt {3} \left (1+\sqrt [3]{2} x\right )}\right )}{\sqrt {3}} \] Input:
Integrate[(2^(2/3) - 2*x)/((2^(2/3) + x)*Sqrt[-1 - x^3]),x]
Output:
(2*2^(2/3)*ArcTanh[Sqrt[-1 - x^3]/(Sqrt[3]*(1 + 2^(1/3)*x))])/Sqrt[3]
Time = 0.41 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2562, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {2^{2/3}-2 x}{\left (x+2^{2/3}\right ) \sqrt {-x^3-1}} \, dx\) |
\(\Big \downarrow \) 2562 |
\(\displaystyle 2\ 2^{2/3} \int \frac {1}{1-\frac {3 \left (\sqrt [3]{2} x+1\right )^2}{-x^3-1}}d\frac {\sqrt [3]{2} x+1}{\sqrt {-x^3-1}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {2\ 2^{2/3} \text {arctanh}\left (\frac {\sqrt {3} \left (\sqrt [3]{2} x+1\right )}{\sqrt {-x^3-1}}\right )}{\sqrt {3}}\) |
Input:
Int[(2^(2/3) - 2*x)/((2^(2/3) + x)*Sqrt[-1 - x^3]),x]
Output:
(2*2^(2/3)*ArcTanh[(Sqrt[3]*(1 + 2^(1/3)*x))/Sqrt[-1 - x^3]])/Sqrt[3]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((e_) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_ Symbol] :> Simp[2*(e/d) Subst[Int[1/(1 + 3*a*x^2), x], x, (1 + 2*d*(x/c)) /Sqrt[a + b*x^3]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] && EqQ[b*c^3 - 4*a*d^3, 0] && EqQ[2*d*e + c*f, 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 2.48 (sec) , antiderivative size = 112, normalized size of antiderivative = 2.87
method | result | size |
trager | \(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-6 \,2^{\frac {1}{3}}\right ) \ln \left (\frac {12 \sqrt {-x^{3}-1}\, x +3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-6 \,2^{\frac {1}{3}}\right ) 2^{\frac {2}{3}} x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-6 \,2^{\frac {1}{3}}\right ) x^{3}+6 \sqrt {-x^{3}-1}\, 2^{\frac {2}{3}}+6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-6 \,2^{\frac {1}{3}}\right ) 2^{\frac {1}{3}} x +2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-6 \,2^{\frac {1}{3}}\right )}{\left (2^{\frac {1}{3}} x +2\right )^{3}}\right )}{3}\) | \(112\) |
default | \(\frac {4 i \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x +1}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}-1}}-\frac {2 i 2^{\frac {2}{3}} \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x +1}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \operatorname {EllipticPi}\left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}+2^{\frac {2}{3}}}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {-x^{3}-1}\, \left (\frac {1}{2}+\frac {i \sqrt {3}}{2}+2^{\frac {2}{3}}\right )}\) | \(249\) |
elliptic | \(\frac {4 i \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x +1}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}-1}}-\frac {2 i 2^{\frac {2}{3}} \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x +1}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \operatorname {EllipticPi}\left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}+2^{\frac {2}{3}}}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {-x^{3}-1}\, \left (\frac {1}{2}+\frac {i \sqrt {3}}{2}+2^{\frac {2}{3}}\right )}\) | \(249\) |
Input:
int((2^(2/3)-2*x)/(2^(2/3)+x)/(-x^3-1)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/3*RootOf(_Z^2-6*2^(1/3))*ln((12*(-x^3-1)^(1/2)*x+3*RootOf(_Z^2-6*2^(1/3) )*2^(2/3)*x^2-RootOf(_Z^2-6*2^(1/3))*x^3+6*(-x^3-1)^(1/2)*2^(2/3)+6*RootOf (_Z^2-6*2^(1/3))*2^(1/3)*x+2*RootOf(_Z^2-6*2^(1/3)))/(2^(1/3)*x+2)^3)
Leaf count of result is larger than twice the leaf count of optimal. 241 vs. \(2 (29) = 58\).
Time = 0.12 (sec) , antiderivative size = 241, normalized size of antiderivative = 6.18 \[ \int \frac {2^{2/3}-2 x}{\left (2^{2/3}+x\right ) \sqrt {-1-x^3}} \, dx=\frac {1}{6} \, \sqrt {6} 2^{\frac {1}{6}} \log \left (\frac {x^{18} - 1440 \, x^{15} + 17400 \, x^{12} + 21056 \, x^{9} - 10368 \, x^{6} - 15360 \, x^{3} - 2 \, \sqrt {6} 2^{\frac {1}{6}} {\left (126 \, x^{14} - 2664 \, x^{11} + 4608 \, x^{5} + 2304 \, x^{2} + 2^{\frac {2}{3}} {\left (x^{16} - 310 \, x^{13} + 2332 \, x^{10} + 2656 \, x^{7} - 256 \, x^{4} - 512 \, x\right )} - 2^{\frac {1}{3}} {\left (17 \, x^{15} - 1058 \, x^{12} + 2528 \, x^{9} + 5408 \, x^{6} + 2560 \, x^{3} + 512\right )}\right )} \sqrt {-x^{3} - 1} - 24 \cdot 2^{\frac {2}{3}} {\left (x^{17} - 121 \, x^{14} + 478 \, x^{11} + 1144 \, x^{8} + 608 \, x^{5} + 64 \, x^{2}\right )} + 48 \cdot 2^{\frac {1}{3}} {\left (5 \, x^{16} - 176 \, x^{13} + 83 \, x^{10} + 680 \, x^{7} + 544 \, x^{4} + 128 \, x\right )} - 2048}{x^{18} + 24 \, x^{15} + 240 \, x^{12} + 1280 \, x^{9} + 3840 \, x^{6} + 6144 \, x^{3} + 4096}\right ) \] Input:
integrate((2^(2/3)-2*x)/(2^(2/3)+x)/(-x^3-1)^(1/2),x, algorithm="fricas")
Output:
1/6*sqrt(6)*2^(1/6)*log((x^18 - 1440*x^15 + 17400*x^12 + 21056*x^9 - 10368 *x^6 - 15360*x^3 - 2*sqrt(6)*2^(1/6)*(126*x^14 - 2664*x^11 + 4608*x^5 + 23 04*x^2 + 2^(2/3)*(x^16 - 310*x^13 + 2332*x^10 + 2656*x^7 - 256*x^4 - 512*x ) - 2^(1/3)*(17*x^15 - 1058*x^12 + 2528*x^9 + 5408*x^6 + 2560*x^3 + 512))* sqrt(-x^3 - 1) - 24*2^(2/3)*(x^17 - 121*x^14 + 478*x^11 + 1144*x^8 + 608*x ^5 + 64*x^2) + 48*2^(1/3)*(5*x^16 - 176*x^13 + 83*x^10 + 680*x^7 + 544*x^4 + 128*x) - 2048)/(x^18 + 24*x^15 + 240*x^12 + 1280*x^9 + 3840*x^6 + 6144* x^3 + 4096))
\[ \int \frac {2^{2/3}-2 x}{\left (2^{2/3}+x\right ) \sqrt {-1-x^3}} \, dx=- \int \left (- \frac {2^{\frac {2}{3}}}{x \sqrt {- x^{3} - 1} + 2^{\frac {2}{3}} \sqrt {- x^{3} - 1}}\right )\, dx - \int \frac {2 x}{x \sqrt {- x^{3} - 1} + 2^{\frac {2}{3}} \sqrt {- x^{3} - 1}}\, dx \] Input:
integrate((2**(2/3)-2*x)/(2**(2/3)+x)/(-x**3-1)**(1/2),x)
Output:
-Integral(-2**(2/3)/(x*sqrt(-x**3 - 1) + 2**(2/3)*sqrt(-x**3 - 1)), x) - I ntegral(2*x/(x*sqrt(-x**3 - 1) + 2**(2/3)*sqrt(-x**3 - 1)), x)
\[ \int \frac {2^{2/3}-2 x}{\left (2^{2/3}+x\right ) \sqrt {-1-x^3}} \, dx=\int { -\frac {2 \, x - 2^{\frac {2}{3}}}{\sqrt {-x^{3} - 1} {\left (x + 2^{\frac {2}{3}}\right )}} \,d x } \] Input:
integrate((2^(2/3)-2*x)/(2^(2/3)+x)/(-x^3-1)^(1/2),x, algorithm="maxima")
Output:
-integrate((2*x - 2^(2/3))/(sqrt(-x^3 - 1)*(x + 2^(2/3))), x)
Exception generated. \[ \int \frac {2^{2/3}-2 x}{\left (2^{2/3}+x\right ) \sqrt {-1-x^3}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((2^(2/3)-2*x)/(2^(2/3)+x)/(-x^3-1)^(1/2),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro unding error%%%{1,[1]%%%} / %%%{%%{[1,0,0]:[1,0,0,-2]%%},[1]%%%} Error: Ba d Argumen
Time = 22.56 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.62 \[ \int \frac {2^{2/3}-2 x}{\left (2^{2/3}+x\right ) \sqrt {-1-x^3}} \, dx=\frac {2^{2/3}\,\sqrt {3}\,\ln \left (\frac {{\left (\sqrt {3}+\sqrt {-x^3-1}+2^{1/3}\,\sqrt {3}\,x\right )}^3\,\left (\sqrt {3}-\sqrt {-x^3-1}+2^{1/3}\,\sqrt {3}\,x\right )}{{\left (x+2^{2/3}\right )}^6}\right )}{3} \] Input:
int(-(2*x - 2^(2/3))/((- x^3 - 1)^(1/2)*(x + 2^(2/3))),x)
Output:
(2^(2/3)*3^(1/2)*log(((3^(1/2) + (- x^3 - 1)^(1/2) + 2^(1/3)*3^(1/2)*x)^3* (3^(1/2) - (- x^3 - 1)^(1/2) + 2^(1/3)*3^(1/2)*x))/(x + 2^(2/3))^6))/3
\[ \int \frac {2^{2/3}-2 x}{\left (2^{2/3}+x\right ) \sqrt {-1-x^3}} \, dx=i \left (-2^{\frac {2}{3}} \left (\int \frac {1}{\sqrt {x^{3}+1}\, 2^{\frac {2}{3}}+\sqrt {x^{3}+1}\, x}d x \right )+2 \left (\int \frac {x}{\sqrt {x^{3}+1}\, 2^{\frac {2}{3}}+\sqrt {x^{3}+1}\, x}d x \right )\right ) \] Input:
int((2^(2/3)-2*x)/(2^(2/3)+x)/(-x^3-1)^(1/2),x)
Output:
i*( - 2**(2/3)*int(1/(sqrt(x**3 + 1)*2**(2/3) + sqrt(x**3 + 1)*x),x) + 2*i nt(x/(sqrt(x**3 + 1)*2**(2/3) + sqrt(x**3 + 1)*x),x))